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arxiv: 2606.24746 · v1 · pith:TWCQUTVOnew · submitted 2026-06-23 · 🪐 quant-ph · cs.CC· cs.IT· math.IT

Asymptotic Compression of Interactive Quantum Communication using Type-Constrained de Finetti Reduction

Louis Desruisseaux , Simon Ducharme , Gurleen Padda , Dave Touchette This is my paper

Pith reviewed 2026-06-26 00:02 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCcs.ITmath.IT
keywords de Finetti reductionquantum interactive communicationinformation costamortized costtype constraintmethod of typesasymptotic compressionpermutation invariance
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The pith

Type-constrained de Finetti reduction proves prior-free quantum information cost equals worst-case amortized quantum communication cost for interactive protocols.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a de Finetti reduction that incorporates the type of the classical input distribution, obtained directly from elementary properties of the method of types. This reduction is applied to the asymptotic compression of quantum interactive communication protocols that receive classical inputs and obey permutation-invariance symmetry. The central result is an equality between the prior-free quantum information cost and the amortized quantum communication cost measured over worst-case inputs. A reader would care because the equality converts worst-case analysis into an i.i.d. problem while preserving the cost relation.

Core claim

Using the type-constrained de Finetti reduction for classical probability distributions, the paper proves that the prior-free quantum information cost equals the worst-case input amortized quantum communication cost for quantum interactive communication protocols with classical inputs.

What carries the argument

The type-constrained de Finetti reduction, which uses the input type to map permutation-invariant worst-case inputs to i.i.d. instances without altering the cost equality.

If this is right

  • Protocol compression rates can be computed via i.i.d. techniques even when inputs are chosen adversarially.
  • The prior-free information cost fully determines the amortized communication cost in the worst-case setting.
  • Proofs of cost equalities in quantum communication become available through direct application of type properties rather than more involved reductions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same type constraint might be adapted to other quantum tasks that admit classical inputs and permutation symmetry.
  • If the method extends beyond classical inputs, it could simplify analysis of fully quantum interactive protocols.

Load-bearing premise

The inputs to the protocols must possess permutation-invariance symmetry so the type-constrained reduction can convert worst-case analysis to i.i.d. analysis while keeping the equality valid.

What would settle it

An explicit interactive protocol with classical inputs satisfying permutation symmetry for which the prior-free quantum information cost differs numerically from the worst-case amortized communication cost would falsify the equality.

Figures

Figures reproduced from arXiv: 2606.24746 by Dave Touchette, Gurleen Padda, Louis Desruisseaux, Simon Ducharme.

Figure 1
Figure 1. Figure 1: Visual representation of the QRST task that has to be simulated on a type-constrained de Finetti [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visual representation of the state redistribution protocol applied to the output de Finetti state [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: State redistribution of a type-constrained de Finetti state with side information at the receiver. [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A general quantum interactive communication protocol Π with classical inputs, consisting of [PITH_FULL_IMAGE:figures/full_fig_p036_4.png] view at source ↗
read the original abstract

For many information processing tasks, de Finetti-style theorems can often simplify the analysis in worst-case input scenarios for which the task exhibits some permutation-invariance symmetry, as they can allow for a reduction from an analysis on worst-case inputs to that of i.i.d. inputs. If further information is available on the inputs, it might be advantageous to reflect this information in the de Finetti reduction. In our work, we focus on a form of such constraint, based on the type of the input. This allows us to obtain a conceptually simple proof of a new de Finetti reduction for classical probability distributions, derived from elementary properties from the method of types. We apply our constrained de Finetti reduction to the compression of quantum interactive communication protocols with classical inputs, and prove that the prior-free quantum information cost equals the worst-case input amortized quantum communication cost.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a type-constrained de Finetti reduction for classical distributions, obtained from elementary method-of-types properties. It applies the reduction to quantum interactive communication protocols with classical inputs and proves that the prior-free quantum information cost equals the worst-case-input amortized quantum communication cost.

Significance. If valid, the result equates two central quantities in asymptotic quantum communication, allowing worst-case analysis to be reduced to the i.i.d. case via input-type symmetry. The elementary derivation from method-of-types properties is a strength, as it avoids additional parameters or ad-hoc constructions.

major comments (1)
  1. [Application to quantum interactive communication (post-abstract)] Application to interactive protocols: the type-constrained reduction maps worst-case inputs to i.i.d. inputs only if permuting the classical input string leaves the sequence of quantum messages and measurements statistically equivalent. The manuscript does not verify that conditional quantum operations (generated round-by-round on prior outcomes) commute with arbitrary input permutations; without this, a non-vanishing discrepancy may survive the asymptotic limit and undermine the claimed equality.
minor comments (1)
  1. The abstract states that the reduction 'follows from elementary method-of-types properties' but does not indicate where the error terms or type-class size bounds are controlled; a short explicit statement of the reduction theorem (with its error scaling) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its significance. We address the single major comment below.

read point-by-point responses

  1. Referee: Application to interactive protocols: the type-constrained reduction maps worst-case inputs to i.i.d. inputs only if permuting the classical input string leaves the sequence of quantum messages and measurements statistically equivalent. The manuscript does not verify that conditional quantum operations (generated round-by-round on prior outcomes) commute with arbitrary input permutations; without this, a non-vanishing discrepancy may survive the asymptotic limit and undermine the claimed equality.

    Authors: We appreciate the referee highlighting this subtlety. The manuscript applies the type-constrained de Finetti reduction to the classical inputs of the interactive protocol; because the inputs are classical, a permutation of the input string induces a corresponding relabeling of the protocol's local classical processing steps. The quantum messages and measurements are generated conditionally on these classical values and prior outcomes, but the overall joint distribution remains invariant under such relabelings when the input type is held fixed, as the protocol treats each input instance symmetrically in the amortized setting. Nevertheless, the manuscript does not contain an explicit verification of this invariance. We will add a short clarifying paragraph in the application section to state this symmetry explicitly and confirm that no non-vanishing discrepancy arises in the asymptotic limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies new reduction to obtain equality

full rationale

The paper first derives a type-constrained de Finetti reduction for classical distributions from elementary method-of-types properties, then applies this reduction to interactive quantum communication protocols under a permutation-invariance assumption on classical inputs. The claimed equality (prior-free QIC equals worst-case amortized QC) is presented as a consequence of this application rather than being presupposed by the definition of either quantity or by any fitted parameter. No equations or steps reduce a prediction to an input by construction, and no load-bearing self-citation chain or smuggled ansatz is exhibited. The derivation remains self-contained against the external symmetry assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard information-theoretic background (method of types and de Finetti theorems) with no free parameters, invented entities, or ad-hoc axioms visible in the abstract.

axioms (1)
  • standard math Elementary properties of the method of types for classical probability distributions
    Invoked to derive the type-constrained de Finetti reduction.

pith-pipeline@v0.9.1-grok · 5690 in / 1275 out tokens · 21629 ms · 2026-06-26T00:02:33.777986+00:00 · methodology

discussion (0)

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Reference graph

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